Temperature dependence of the internal dynamics of a calmodulin-peptide complex

Deuterium spin relaxation of fractionally deuterated ribonuclease H using paired 475 and 950 MHz NMR spectrometers

Deuterium (2H) spin relaxation of 13CH2D methyl groups has been widely applied to investigate picosecond-to-nanosecond conformational dynamics in proteins by solution-state NMR spectroscopy. The B0 dependence of the 2H spin relaxation rates is represented by a linear relationship between the spectral density function at three discrete frequencies J(0), J(ωD) and J(2ωD). In this study, the linear relation between 2H relaxation rates at B0 fields separated by a factor of two and the interpolation of rates at intermediate frequencies are combined for a more robust approach for spectral density mapping. The general usefulness of the approach is demonstrated on a fractionally deuterated (55%) and alternate 13C-12C labeled sample of E. coli RNase H. Deuterium relaxation rate constants (R1, R1ρ, RQ, RAP) were measured for 57 well-resolved 13CH2D moieties in RNase H at 1H frequencies of 475 MHz, 500 MHz, 900 MHz, and 950 MHz. The spectral density mapping of the 475/950 MHz data combination was performed independently and jointly to validate the expected relationship between data recorded at B0 fields separated by a factor of two. The final analysis was performed by jointly analyzing 475/950 MHz rates with 700 MHz rates interpolated from 500/900 MHz data to yield six J(ωD) values for each methyl peak. The J(ω) profile for each peak was fit to the original (τM, Sf2, τf) or extended model-free function (τM, Sf2, Ss2, τf, τs) to obtain optimized dynamic parameters.

Deep mining of the protein energy landscape

For over half a century, it has been known that protein molecules naturally undergo extensive structural fluctuations, and that these internal motions are intimately related to their functional properties. The energy landscape view has provided a powerful framework for describing the various physical states that proteins visit during their lifetimes. This Perspective focuses on the commonly neglected and often disparaged axis of the protein energy landscape: entropy. Initially seen largely as a barrier to functionally relevant states of protein molecules, it has recently become clear that proteins retain considerable conformational entropy in the "native" state, and that this entropy can and often does contribute significantly to the free energy of fundamental protein properties, processes, and functions. NMR spectroscopy, molecular dynamics simulations, and emerging crystallographic views have matured in parallel to illuminate dynamic disorder of the "ground state" of proteins and their importance in not only transiting between biologically interesting structures but also greatly influencing their stability, cooperativity, and contribution to critical properties such as allostery.

Allostery in the dynamic coactivator domain KIX occurs through minor conformational micro-states

The coactivator KIX of CBP uses two binding surfaces to recognize multiple activators and exhibits allostery in ternary complex formation. Activator•coactivator interactions are central to transcriptional regulation, yet the microscopic origins of allostery in dynamic proteins like KIX are largely unknown. Here, we investigate the molecular recognition and allosteric manifestations involved in two KIX ternary systems c-Myb•KIX•MLL and pKID•KIX•MLL. Exploring the hypothesis that binary complex formation prepays an entropic cost for positive cooperativity, we utilize molecular dynamics simulations, side chain methyl order parameters, and differential scanning fluorimetry (DSF) to explore conformational entropy changes in KIX. The protein's configurational micro-states from structural clustering highlight the utility of protein plasticity in molecular recognition and allostery. We find that apo KIX occupies a wide distribution of lowly-populated configurational states. Each binding partner has its own suite of KIX states that it selects, building a model of molecular recognition fingerprints. Allostery is maximized with MLL pre-binding, which corresponds to the observation of a significant reduction in KIX micro-states observed when MLL binds. With all binding partners, the changes in KIX conformational entropy arise predominantly from changes in the most flexible loop. Likewise, we find that a small molecule and mutations allosterically inhibit/enhance activator binding by tuning loop dynamics, suggesting that loop-targeting chemical probes could be developed to alter KIX•activator interactions. Experimentally capturing KIX stabilization is challenging, particularly because of the disordered nature of particular activators. However, DSF melting curves allow for inference of relative entropic changes that occur across complexes, which we compare to our computed entropy changes using simulation methyl order parameters.

Correlated Response of Protein Side-Chain Fluctuations and Conformational Entropy to Ligand Binding

The heterogeneous fast side-chain dynamics of proteins plays crucial roles in molecular recognition and binding. Site-specific NMR experiments quantify these motions by measuring the model-free order parameter (O_axis²) on a scale of 0 (most flexible) to 1 (least flexible) for each methyl-containing residue of proteins. Here, we have examined ligand-induced variations in the fast side-chain dynamics and conformational entropy of calmodulin (CaM) using five different CaM-peptide complexes. O_axis² of CaM in the ligand-free (O_axis,U²) and ligand-bound (O_axis,B²) states are calculated from molecular dynamics trajectories and conformational energy surfaces obtained using the adaptive biasing force (ABF) method. ΔO_axis² = O_axis,B² - O_axis,U² follows a Gaussian-like unimodal distribution whose second moment is a potential indicator of the binding affinity of these complexes. The probability for the binding-induced O_axis,U² → O_axis,B² transition decreases with increasing magnitude of ΔO_axis², indicating that large flexibility changes are improbable for side chains of CaM after ligand binding. A linear correlation established between ΔO_axis² and the conformational entropy change of the protein makes possible the determination of the conformational entropy of binding of protein-ligand complexes. The results not only underscore the functional importance of fast side-chain fluctuations but also highlight key motional and thermodynamic correlates of protein-ligand binding.

An Integrative Approach to Determine 3D Protein Structures Using Sparse Paramagnetic NMR Data and Physical Modeling

Paramagnetic nuclear magnetic resonance (NMR) methods have emerged as powerful tools for structure determination of large, sparsely protonated proteins. However traditional applications face several challenges, including a need for large datasets to offset the sparsity of restraints, the difficulty in accounting for the conformational heterogeneity of the spin-label, and noisy experimental data. Here we propose an integrative approach to structure determination combining sparse paramagnetic NMR with physical modelling to infer approximate protein structural ensembles. We use calmodulin in complex with the smooth muscle myosin light chain kinase peptide as a model system. Despite acquiring data from samples labeled only at the backbone amide positions, we are able to produce an ensemble with an average RMSD of ∼2.8 Å from a reference X-ray crystal structure. Our approach requires only backbone chemical shifts and measurements of the paramagnetic relaxation enhancement and residual dipolar couplings that can be obtained from sparsely labeled samples.

On Catalytic Kinetics of Enzymes

Classical enzyme kinetic theories are summarized and linked with modern discoveries here. The sequential catalytic events along time axis by enzyme are analyzed at the molecular level, and by using master equations, this writing tries to connect the microscopic molecular behavior of enzyme to kinetic data (like velocity and catalytic coefficient k) obtained in experiment: 1/k = t equals to the sum of the times taken by the constituent individual steps. The relationships between catalytic coefficient k, catalytic rate or velocity, the amount of time taken by each step and physical or biochemical conditions of the system are discussed, and the perspective and hypothetic equations proposed here regarding diffusion, conformational change, chemical conversion, product release steps and the whole catalytic cycle provide an interpretation of previous experimental observations and can be testified by future experiments.

Local ordering and dynamics in anisotropic media by magnetic resonance: from liquid crystals to proteins

Magnetic resonance methods have been used extensively for over 50 years to elucidate molecular structure and dynamics of liquid crystals (LCs), providing information quite unique in its rigour and extent. The ESR- or NMR-active probe is often a solute molecule reporting on characteristics associated with the surrounding (LC) medium, which exerts the spatial restrictions on the probe. The theoretical approaches developed for LCs are applicable to anisotropic media in general. Of particular interest is the interior space of a globular protein labelled, e.g. with a nitroxide moiety or a ¹⁵N-¹H bond. The ESR or NMR label plays the role of the probe and the internal protein surroundings the role of the anisotropic medium. A general feature of the restricted motions is the local ordering, i.e. the nature, magnitude and symmetry of the spatial restraints exerted at the site of the moving probe. This property is the main theme of the present review article. We outline its treatment in our work from both the theoretical and the experimental points of view, highlighting the new physical insights gained. Our illustrations include studies on thermotropic (nematic and smectic) and lyotropic liquid crystals formed by phospholipids, in addition to studies of proteins.

Characterization of Internal Protein Dynamics and Conformational Entropy by NMR Relaxation

Recent studies suggest that the fast timescale motion of methyl-bearing side chains may play an important role in mediating protein activity. These motions have been shown to encapsulate the residual conformational entropy of the folded state that can potentially contribute to the energetics of protein function. Here, we provide an overview of how to characterize these motions using nuclear magnetic resonance (NMR) spin relaxation methods. The strengths and limitations of several techniques are highlighted in order to assist with experimental design. Particular emphasis is placed on the practical aspects of sample preparation, data collection, data fitting, and statistical analysis. Additionally, discussion of the recently refined "entropy meter" is presented and its use in converting NMR observables to conformational entropy is illustrated. Taken together, these methods should yield new insights into the complex interplay between structure and dynamics in protein function.

Companion Simulations and Modeling to NMR-Based Dynamical Studies of Proteins

NMR-based studies of protein dynamics and molecular simulations have a synergistic relationship. Molecular simulations, in combination with interpretative theoretical models, leverage the dynamical information obtained from NMR. They provide the concrete physical schema underlying the quantities measured by NMR, and help extend the range of applications beyond the strictly dynamic properties. NMR data in turn provide concrete data to test and improve the potential functions used for simulation of dynamics of proteins. The concept of time correlation functions is central to the understanding of many dynamical processes. Their evaluation through atomistic simulations is discussed, with application to different dynamical quantities measured by NMR. While advances in computers have made such atomistic simulations almost routine, the companion use of simple interpretive models is stressed, to provide not just numbers but physical insight.

Practical aspects of high-pressure NMR spectroscopy and its applications in protein biophysics and structural biology

Pressure and temperature are the two fundamental variables of thermodynamics. Temperature and chemical perturbation are central experimental tools for the exploration of macromolecular structure and dynamics. Though it has long been recognized that hydrostatic pressure offers a complementary and often unique view of macromolecular structure, stability and dynamics, it has not been employed nearly as much. For solution NMR applications the limited use of high-pressure is undoubtedly traced to difficulties of employing pressure in the context of modern multinuclear and multidimensional NMR. Limitations in pressure tolerant NMR sample cells have been overcome and enable detailed studies of macromolecular energy landscapes, hydration, dynamics and function. Here we review the practical considerations for studies of biological macromolecules at elevated pressure, with a particular emphasis on applications in protein biophysics and structural biology.

Functional Role of Solvent Entropy and Conformational Entropy of Metal Binding in a Dynamically Driven Allosteric System

Allostery is a regulatory phenomenon whereby ligand binding to one site influences the binding of the same or a different ligand to another site on a macromolecule. The physical origins of allosteric regulation remain under intense investigation. In general terms, ligand-induced structural changes, perturbations of residue-specific dynamics, and surrounding solvent molecules all potentially contribute to the global energetics of allostery. While the role of solvent is generally well understood in regulatory events associated with major protein structural rearrangements, the degree to which protein dynamics impact solvent degrees of freedom is unclear, particularly in cases of dynamically driven allostery. With the aid of new crystal structures, extensive calorimetric and residue-specific dynamics studies over a range of time scales and temperatures, we dissect for the first time the relative degree to which changes in solvent entropy and residue-specific dynamics impact dynamically driven, allosteric inhibition of DNA binding by Zn in the zinc efflux repressor, CzrA (chromosomal zinc-regulated repressor). We show that non-native residue-specific dynamics in allosterically impaired CzrA mutants are accompanied by significant perturbations in solvent entropy that cannot be predicted from crystal structures. We conclude that functional dynamics are not necessarily restricted to protein residues but involve surface water molecules that may be responding to ligand (Zn)-mediated perturbations in protein internal motions that define the conformational ensemble, rather than major structural rearrangements.

Self-Consistent Framework Connecting Experimental Proxies of Protein Dynamics with Configurational Entropy

The recently developed NMR techniques enable estimation of protein configurational entropy change from the change in the average methyl order parameters. This experimental observable, however, does not directly measure the contribution of intramolecular couplings, protein main-chain motions, or angular dynamics. Here, we carry out a self-consistent computational analysis of the impact of these missing contributions on an extensive set of molecular dynamics simulations of different proteins undergoing binding. Specifically, we compare the configurational entropy change in protein complex formation as obtained by the maximum information spanning tree approximation (MIST), which treats the above entropy contributions directly, and the change in the average NMR methyl and NH order parameters. Our parallel implementation of MIST allows us to treat hard angular degrees of freedom as well as couplings up to full pairwise order explicitly, while still involving a high degree of sampling and tackling molecules of biologically relevant sizes. First, we demonstrate a remarkably strong linear relationship between the total configurational entropy change and the average change in both methyl and backbone-NH order parameters. Second, in contrast to canonical assumptions, we show that the main-chain and angular terms contribute significantly to the overall configurational entropy change and also scale linearly with it. Consequently, linear models starting from the average methyl order parameters are able to capture the contribution of main-chain and angular terms well. After applying the quantum-mechanical harmonic oscillator entropy formalism, we establish a similarly strong linear relationship for X-ray crystallographic B-factors. Finally, we demonstrate that the observed linear relationships remain robust against drastic undersampling and argue that they reflect an intrinsic property of compact proteins. Despite their remarkable strength, however, the above linear relationships yield estimates of configurational entropy change whose accuracy appears to be sufficient for qualitative applications only.

Measuring Entropy in Molecular Recognition by Proteins

Molecular recognition by proteins is fundamental to the molecular basis of biology. Dissection of the thermodynamic landscape governing protein-ligand interactions has proven difficult because determination of various entropic contributions is quite challenging. Nuclear magnetic resonance relaxation measurements, theory, and simulations suggest that conformational entropy can be accessed through a dynamical proxy. Here, we review the relationship between measures of fast side-chain motion and the underlying conformational entropy. The dynamical proxy reveals that the contribution of conformational entropy can range from highly favorable to highly unfavorable and demonstrates the potential of this key thermodynamic variable to modulate protein-ligand interactions. The dynamical so-called entropy meter also refines the role of solvent entropy and directly determines the loss in rotational-translational entropy that occurs upon formation of high-affinity complexes. The ability to quantify the roles of entropy through an entropy meter based on measurable dynamical properties promises to highlight its role in protein function.

Entropy in molecular recognition by proteins

Molecular recognition by proteins is fundamental to molecular biology. Dissection of the thermodynamic energy terms governing protein-ligand interactions has proven difficult, with determination of entropic contributions being particularly elusive. NMR relaxation measurements have suggested that changes in protein conformational entropy can be quantitatively obtained through a dynamical proxy, but the generality of this relationship has not been shown. Twenty-eight protein-ligand complexes are used to show a quantitative relationship between measures of fast side-chain motion and the underlying conformational entropy. We find that the contribution of conformational entropy can range from favorable to unfavorable, which demonstrates the potential of this thermodynamic variable to modulate protein-ligand interactions. For about one-quarter of these complexes, the absence of conformational entropy would render the resulting affinity biologically meaningless. The dynamical proxy for conformational entropy or "entropy meter" also allows for refinement of the contributions of solvent entropy and the loss in rotational-translational entropy accompanying formation of high-affinity complexes. Furthermore, structure-based application of the approach can also provide insight into long-lived specific water-protein interactions that escape the generic treatments of solvent entropy based simply on changes in accessible surface area. These results provide a comprehensive and unified view of the general role of entropy in high-affinity molecular recognition by proteins.

Entropy redistribution controls allostery in a metalloregulatory protein

Allosteric communication between two ligand-binding sites in a protein is a central aspect of biological regulation that remains mechanistically unclear. Here we show that perturbations in equilibrium picosecond-nanosecond motions impact zinc (Zn)-induced allosteric inhibition of DNA binding by the Zn efflux repressor CzrA (chromosomal zinc-regulated repressor). DNA binding leads to an unanticipated increase in methyl side-chain flexibility and thus stabilizes the complex entropically; Zn binding redistributes these motions, inhibiting formation of the DNA complex by restricting coupled fast motions and concerted slower motions. Allosterically impaired CzrA mutants are characterized by distinct nonnative fast internal dynamics "fingerprints" upon Zn binding, and DNA binding is weakly regulated. We demonstrate the predictive power of the wild-type dynamics fingerprint to identify key residues in dynamics-driven allostery. We propose that driving forces arising from dynamics can be harnessed by nature to evolve new allosteric ligand specificities in a compact molecular scaffold.

Hierarchical molecular dynamics of bovine serum albumin in concentrated aqueous solution below and above thermal denaturation

The dynamics of proteins in solution is a complex and hierarchical process, affected by the aqueous environment as well as temperature. We present a comprehensive study on nanosecond time and nanometer length scales below, at, and above the denaturation temperature Td. Our experimental data evidence dynamical processes in protein solutions on three distinct time scales. We suggest a consistent physical picture of hierarchical protein dynamics: (i) self-diffusion of the entire protein molecule is confirmed to agree with colloid theory for all temperatures where the protein is in its native conformational state. At higher temperatures T > Td, the self-diffusion is strongly obstructed by cross-linking or entanglement. (ii) The amplitude of backbone fluctuations grows with increasing T, and a transition in its dynamics is observed above Td. (iii) The number of mobile side-chains increases sharply at Td while their average dynamics exhibits only little variations. The combination of quasi-elastic neutron scattering and the presented analytical framework provides a detailed microscopic picture of the protein molecular dynamics in solution, thereby reflecting the changes of macroscopic properties such as cluster formation and gelation.

The unusual internal motion of the villin headpiece subdomain

The thermostable 36-residue subdomain of the villin headpiece (HP36) is the smallest known cooperatively folding protein. Although the folding and internal dynamics of HP36 and close variants have been extensively studied, there has not been a comprehensive investigation of side-chain motion in this protein. Here, the fast motion of methyl-bearing amino acid side chains is explored over a range of temperatures using site-resolved solution nuclear magnetic resonance deuterium relaxation. The squared generalized order parameters of methyl groups extensively spatially segregate according to motional classes. This has not been observed before in any protein studied using this methodology. The class segregation is preserved from 275 to 305 K. Motions detected in Helix 3 suggest a fast timescale of conformational heterogeneity that has not been previously observed but is consistent with a range of folding and dynamics studies. Finally, a comparison between the order parameters in solution with previous results based on solid-state nuclear magnetic resonance deuterium line shape analysis of HP36 in partially hydrated powders shows a clear disagreement for half of the sites. This result has significant implications for the interpretation of data derived from a variety of approaches that rely on partially hydrated protein samples.

Unveiling Inherent Degeneracies in Determining Population-Weighted Ensembles of Interdomain Orientational Distributions Using NMR Residual Dipolar Couplings: Application to RNA Helix Junction Helix Motifs

A growing number of studies employ time-averaged experimental data to determine dynamic ensembles of biomolecules. While it is well-known that different ensembles can satisfy experimental data to within error, the extent and nature of these degeneracies, and their impact on the accuracy of the ensemble determination remains poorly understood. Here, we use simulations and a recently introduced metric for assessing ensemble similarity to explore degeneracies in determining ensembles using NMR residual dipolar couplings (RDCs) with specific application to A-form helices in RNA. Various target ensembles were constructed representing different domain-domain orientational distributions that are confined to a topologically restricted (<10%) conformational space. Five independent sets of ensemble averaged RDCs were then computed for each target ensemble and a "sample and select" scheme used to identify degenerate ensembles that satisfy RDCs to within experimental uncertainty. We find that ensembles with different ensemble sizes and that can differ significantly from the target ensemble (by as much as ∑Ω ∼ 0.4 where ∑Ω varies between 0 and 1 for maximum and minimum ensemble similarity, respectively) can satisfy the ensemble averaged RDCs. These deviations increase with the number of unique conformers and breadth of the target distribution, and result in significant uncertainty in determining conformational entropy (as large as 5 kcal/mol at T = 298 K). Nevertheless, the RDC-degenerate ensembles are biased toward populated regions of the target ensemble, and capture other essential features of the distribution, including the shape. Our results identify ensemble size as a major source of uncertainty in determining ensembles and suggest that NMR interactions such as RDCs and spin relaxation, on their own, do not carry the necessary information needed to determine conformational entropy at a useful level of precision. The framework introduced here provides a general approach for exploring degeneracies in ensemble determination for different types of experimental data.

Fluctuations within folded proteins: implications for thermodynamic and allosteric regulation

Folded protein structures are both stable and dynamic. Historically, our clearest window into these structures came from X-ray crystallography, which generally provided a static image of each protein's singular "folded state", highlighting its stability. Deviations away from that crystallographic structure were difficult to quantify, and as a result, their potential functional consequences were often neglected. However, several dynamical and statistical studies now highlight the structural variability that is present within the protein's folded state. Here we review mounting evidence of the importance of these structural rearrangements; both experiment and computation indicate that folded proteins undergo substantial fluctuations that can greatly influence their function. Crucially, recent studies have shown that structural elements of proteins, especially their side-chain degrees of freedom, fluctuate in ways that generate significant conformational heterogeneity. The entropy associated with these motions contributes to the folded structure's thermodynamic stability. In addition, since these fluctuations can shift in response to perturbations such as ligand binding, they may play an important role in the protein's capacity to respond to environmental cues. In one compelling example, the entropy associated with side-chain fluctuations contributes significantly to regulating the binding of calmodulin to a set of peptide ligands. The neglect of fluctuations within proteins' native states was often justified by the dense packing within folded proteins, which has inspired comparisons with crystalline solids. Many liquids, however, can achieve similarly dense packing yet fluidity is maintained through correlated molecular motions. Indeed, the studies we discuss favor comparison of folded proteins not with solids but instead with dense liquids, where the internal side chain fluidity is facilitated by collective motions that are correlated over long distances. These correlated rearrangements can enable allosteric communication between different parts of a protein, through subtle and varied channels. Such long-range correlations appear to be an innate feature of proteins in general, manifest even in molecules lacking known allosteric regulators and arising robustly from the physical nature of their internal environment. Given their ubiquity, it is only to be expected that, over time, nature has refined some subset of these correlated motions and put them to use. Native state fluctuations increasingly appear to be vital for proteins' natural functions. Understanding the diversity, origin, and range of these rearrangements may provide novel routes for rationally manipulating biomolecular activity.

On the relationship between NMR-derived amide order parameters and protein backbone entropy changes

Molecular dynamics simulations are used to analyze the relationship between NMR-derived squared generalized order parameters of amide NH groups and backbone entropy. Amide order parameters (O(2) NH ) are largely determined by the secondary structure and average values appear unrelated to the overall flexibility of the protein. However, analysis of the more flexible subset (O(2) NH  < 0.8) shows that these report both on the local flexibility of the protein and on a different component of the conformational entropy than that reported by the side chain methyl axis order parameters, O(2) axis . A calibration curve for backbone entropy vs. O(2) NH is developed, which accounts for both correlations between amide group motions of different residues, and correlations between backbone and side chain motions. This calibration curve can be used with experimental values of O(2) NH changes obtained by NMR relaxation measurements to extract backbone entropy changes, for example, upon ligand binding. In conjunction with our previous calibration for side chain entropy derived from measured O(2) axis values this provides a prescription for determination of the total protein conformational entropy changes from NMR relaxation measurements.

Temperature dependence of internal motions of protein side-chain NH3(+) groups: insight into energy barriers for transient breakage of hydrogen bonds

Although charged side chains play important roles in protein function, their dynamic properties are not well understood. Nuclear magnetic resonance methods for investigating the dynamics of lysine side-chain NH3(+) groups were established recently. Using this methodology, we have studied the temperature dependence of the internal motions of the lysine side-chain NH3(+) groups that form ion pairs with DNA phosphate groups in the HoxD9 homeodomain-DNA complex. For these NH3(+) groups, we determined order parameters and correlation times for bond rotations and reorientations at 15, 22, 28, and 35 °C. The order parameters were found to be virtually constant in this temperature range. In contrast, the bond-rotation correlation times of the NH3(+) groups were found to depend strongly on temperature. On the basis of transition state theory, the energy barriers for NH3(+) rotations were analyzed and compared to those for CH3 rotations. Enthalpies of activation for NH3(+) rotations were found to be significantly higher than those for CH3 rotations, which can be attributed to the requirement of hydrogen bond breakage. However, entropies of activation substantially reduce the overall free energies of activation for NH3(+) rotations to a level comparable to those for CH3 rotations. This entropic reduction in energy barriers may accelerate molecular processes requiring hydrogen bond breakage and play a kinetically important role in protein function.

Resolving Internal Motional Correlations to Complete the Conformational Entropy Meter

Conformational entropy (SΩ) has long been used to theoretically characterize the dynamics of proteins, DNA, and other polymers. Though recent advances enabled its calculation also from simulations and nuclear magnetic resonance (NMR) relaxation experiments, correlated molecular motion has hitherto greatly hindered both numerical and experimental determination, requiring demanding empirical and computational calibrations. Herein, we show that these motional correlations can be estimated directly from the temperature-dependent SΩ series that reveal effective persistence lengths of the polymers, which we demonstrate by measuring SΩ of amphiphilic molecules in model lipid systems by spin-labeling electron paramagnetic resonance (EPR) spectroscopy. We validate our correlation-corrected SΩ meter against the basic biophysical interactions underlying biomembrane formation and stability, against the changes in enthalpy and diffusion coefficients upon phase transitions, and against the energetics of fatty acid dissociation. As the method can be directly applied to conformational analysis of proteins and other polymers, as well as adapted to NMR or polarized fluorescence techniques, we believe that the approach can greatly enrich the scope of experimentally available statistical thermodynamics, offering new physical insights into the behavior of biomolecules.

Banding of NMR-derived methyl order parameters: implications for protein dynamics

Our understanding of protein folding, stability, and function has begun to more explicitly incorporate dynamical aspects. Nuclear magnetic resonance has emerged as a powerful experimental method for obtaining comprehensive site-resolved insight into protein motion. It has been observed that methyl-group motion tends to cluster into three "classes" when expressed in terms of the popular Lipari-Szabo model-free squared generalized order parameter. Here the origins of the three classes or bands in the distribution of order parameters are examined. As a first step, a Bayesian based approach, which makes no a priori assumption about the existence or number of bands, is developed to detect the banding of Oaxis2 values derived either from NMR experiments or molecular dynamics simulations. The analysis is applied to seven proteins with extensive molecular dynamics simulations of these proteins in explicit water to examine the relationship between O2 and fine details of the motion of methyl bearing side chains. All of the proteins studied display banding, with some subtle differences. We propose a very simple yet plausible physical mechanism for banding. Finally, our Bayesian method is used to analyze the measured distributions of methyl group motions in the catabolite activating protein and several of its mutants in various liganded states and discuss the functional implications of the observed banding to protein dynamics and function.

Monitoring side-chain dynamics of proteins using (2)H relaxation

Nuclear magnetic resonance (NMR) is a powerful technique capable of monitoring a wide range of motions in proteins on a per residue basis. A variety of (2)H relaxation experiments have been developed for monitoring side-chain methyl group motions on the picosecond-nanosecond timescale. These experiments enable determination of the order parameter, S (2) axis, which reports on the rigidity of the C-CH3 bond for side-chain methyl groups. The application of a commonly used subset of these experiments is described in this chapter. It is intended to serve as a practical guide to investigators interested in monitoring side-chain motions.

Microscopic insights into the NMR relaxation-based protein conformational entropy meter

Conformational entropy is a potentially important thermodynamic parameter contributing to protein function. Quantitative measures of conformational entropy are necessary for an understanding of its role but have been difficult to obtain. An empirical method that utilizes changes in conformational dynamics as a proxy for changes in conformational entropy has recently been introduced. Here we probe the microscopic origins of the link between conformational dynamics and conformational entropy using molecular dynamics simulations. Simulation of seven proteins gave an excellent correlation with measures of side-chain motion derived from NMR relaxation. The simulations show that the motion of methyl-bearing side chains are sufficiently coupled to that of other side chains to serve as excellent reporters of the overall side-chain conformational entropy. These results tend to validate the use of experimentally accessible measures of methyl motion--the NMR-derived generalized order parameters--as a proxy from which to derive changes in protein conformational entropy.

The time correlation function perspective of NMR relaxation in proteins

We applied over a decade ago the two-body coupled-rotator slowly relaxing local structure (SRLS) approach to NMR relaxation in proteins. One rotator is the globally moving protein and the other rotator is the locally moving probe (spin-bearing moiety, typically the (15)N-(1)H bond). So far we applied SRLS to (15)N-H relaxation from seven different proteins within the scope of the commonly used data-fitting paradigm. Here, we solve the SRLS Smoluchowski equation using typical best-fit parameters as input, to obtain the corresponding generic time correlation functions (TCFs). The following new information is obtained. For actual rhombic local ordering and main ordering axis pointing along C(i-1)(α)-C(i)(α), the measurable TCF is dominated by the (K,K') = (-2,2), (2,2), and (0,2) components (K is the order of the rank 2 local ordering tensor), determined largely by the local motion. Global diffusion axiality affects the analysis significantly when the ratio between the parallel and perpendicular components exceeds approximately 1.5. Local diffusion axiality has a large and intricate effect on the analysis. Mode-coupling becomes important when the ratio between the global and local motional rates falls below 0.01. The traditional method of analysis--model-free (MF)--represents a simple limit of SRLS. The conditions under which the MF and SRLS TCFs are the same are specified. The validity ranges of wobble-in-a-cone and rotation on the surface of a cone as local motions are determined. The evolution of the intricate Smoluchowski operator from the simple diffusion operator for a sphere reorienting in isotropic medium is delineated. This highlights the fact that SRLS is an extension of the established stochastic theories for treating restricted motions. This study lays the groundwork for TCF-based comparison between mesoscopic SRLS and atomistic molecular dynamics.

A surprising role for conformational entropy in protein function

Formation of high-affinity complexes is critical for the majority of enzymatic reactions involving proteins. The creation of the family of Michaelis and other intermediate complexes during catalysis clearly involves a complicated manifold of interactions that are diverse and complex. Indeed, computing the energetics of interactions between proteins and small molecule ligands using molecular structure alone remains a great challenge. One of the most difficult contributions to the free energy of protein-ligand complexes to access experimentally is that due to changes in protein conformational entropy. Fortunately, recent advances in solution nuclear magnetic resonance (NMR) relaxation methods have enabled the use of measures-of-motion between conformational states of a protein as a proxy for conformational entropy. This review briefly summarizes the experimental approaches currently employed to characterize fast internal motion in proteins, how this information is used to gain insight into conformational entropy, what has been learned, and what the future may hold for this emerging view of protein function.

The dark energy of proteins comes to light: conformational entropy and its role in protein function revealed by NMR relaxation

Historically it has been virtually impossible to experimentally determine the contribution of residual protein entropy to fundamental protein activities such as the binding of ligands. Recent progress has illuminated the possibility of employing NMR relaxation methods to quantitatively determine the role of changes in conformational entropy in molecular recognition by proteins. The method rests on using fast internal protein dynamics as a proxy. Initial results reveal a large and variable role for conformational entropy in the binding of ligands by proteins. Such a role for conformational entropy in molecular recognition has significant implications for enzymology, signal transduction, allosteric regulation and the development of protein-directed pharmaceuticals.

Coupled motion in proteins revealed by pressure perturbation

The cooperative nature of protein substructure and internal motion is a critical aspect of their functional competence about which little is known experimentally. NMR relaxation is used here to monitor the effects of high pressure on fast internal motion in the protein ubiquitin. In contrast to the main chain, the motions of the methyl-bearing side chains have a large and variable pressure dependence. Within the core, this pressure sensitivity correlates with the magnitude of motion at ambient pressure. Spatial clustering of the dynamic response to applied hydrostatic pressure is also seen, indicating localized cooperativity of motion on the sub-nanosecond time scale and suggesting regions of variable compressibility. These and other features indicate that the native ensemble contains a significant fraction of members with characteristics ascribed to the recently postulated "dry molten globule". The accompanying variable side-chain conformational entropy helps complete our view of the thermodynamic architecture underlying protein stability, folding, and function.

Thermal coefficients of the methyl groups within ubiquitin

Physiological processes such as protein folding and molecular recognition are intricately linked to their dynamic signature, which is reflected in their thermal coefficient. In addition, the local conformational entropy is directly related to the degrees of freedom, which each residue possesses within its conformational space. Therefore, the temperature dependence of the local conformational entropy may provide insight into understanding how local dynamics may affect the stability of proteins. Here, we analyze the temperature dependence of internal methyl group dynamics derived from the cross-correlated relaxation between dipolar couplings of two CH bonds within ubiquitin. Spanning a temperature range from 275 to 308 K, internal methyl group dynamics tend to increase with increasing temperature, which translates to a general increase in local conformational entropy. With this data measured over multiple temperatures, the thermal coefficient of the methyl group order parameter, the characteristic thermal coefficient, and the local heat capacity were obtained. By analyzing the distribution of methyl group thermal coefficients within ubiquitin, we found that the N-terminal region has relatively high thermostability. These results indicate that methyl groups contribute quite appreciably to the total heat capacity of ubiquitin through the regulation of local conformational entropy.

Temperature dependence of molecular interactions involved in defining stability of glutamine binding protein and its complex with L-glutamine

The temperature dependence of dynamic parameters derived from nuclear magnetic resonance (NMR) relaxation data is related to conformational entropy of the system under study. This provides information such as macromolecules stability and thermodynamics of ligand binding. We studied the temperature dependence of NMR order parameter of glutamine binding protein (GlnBP), a periplasmic binding protein (PBP) highly specific to L-glutamine associated with its ABC transporter, with the goal of elucidating the dynamical differences between the respective ligand bound and free forms. We found that the protein-ligand interaction, which is stabilized at higher temperature, has a striking effect on the stability of the hydrophobic core of the large domain of GlnBP. Moreover, in contrast to what was found for less specific PBPs, the decreasing backbone motion of the hinge region at increasing temperature supports the idea that the likelihood that GlnBP can adopt a ligand free closed conformation in solution diminishes at higher temperatures. Our results support the induced-fit model as mode of action for GlnBP. In addition, we found that the backbones of residues involved in a salt bridge do not necessarily become more rigid as the temperature rises as it was previously suggested [Vinther, J. M., et al. (2011) J. Am. Chem. Soc., 133, 271-278]. Our results show that for this to happen these residues have to also directly interact with a region of the protein that is becoming more rigid as the temperature increases.

Probing side-chain dynamics in proteins by the measurement of nine deuterium relaxation rates per methyl group

We demonstrate the feasibility of the measurement of up to nine deuterium spin relaxation rates in 13CHD2 and 13CH2D methyl isotopomers of small proteins. In addition to five measurable 2H relaxation rates in a 13CH2D methyl group (Millet, O.; Muhandiram, D. R.; Skrynnikov, N. R.; Kay, L. E. J. Am. Chem. Soc. 2002, 124, 6439-48), the measurement of additional four rates of (nearly) single-exponentially decaying magnetization terms in methyl groups of the 13CHD2 variety is reported. Consistency relationships between 2H spin relaxation rates measured in the two different types of methyl groups are derived and verified experimentally for a subset of methyl-containing side chains in the protein ubiquitin. A detailed comparison of methyl-bearing side-chain dynamics parameters obtained from relaxation measurements in 13CH2D and 13CHD2 methyls of ubiquitin at 10, 27, and 40 °C reveals that transverse 2H relaxation rates in 13CHD2 groups are reliable and accurate reporters of the amplitudes of methyl 3-fold axis motions (S(axis)2) for protein molecules with global molecular tumbling times τ(C) >~9 ns. For smaller molecules, simple correction of transverse 2H relaxation rates in 13CHD2 groups is sufficient for the derivation of robust measures of order. Residue-specific distributions of S(axis)2 are consistent with atomic-detail molecular dynamics (MD) results. Both 13CHD2- and 13CH2D-derived S(axis)2 values are in good overall agreement with those obtained from 1 μs MD simulations at all the three temperatures, although some differences in the site-specific temperature dependence between MD- and 2H-relaxation-derived S(axis)2 values are observed.

Methyl dynamics of a Ca2+-calmodulin-peptide complex from NMR/SRLS

We developed the slowly relaxing local structure (SRLS) approach for analyzing NMR spin relaxation in proteins. SRLS accounts for dynamical coupling between the tumbling of the protein and the local motion of the probe and for general tensorial properties. It is the generalization of the traditional model-free (MF) method, which does not account for mode-coupling and treats only simple tensorial properties. SRLS is applied herein to ²H relaxation of ¹³CDH₂ groups in the complex of Ca(2+)-calmodulin with the peptide smMLCKp. Literature data comprising ²H T₁ and T₂ acquired at 14.1 and 17.6 T, and 288, 295, 308, and 320 K, are used. We find that mode-coupling is a small effect for methyl dynamics. On the other hand, general tensorial properties are important. In particular, it is important to allow for the asymmetry of the local spatial restrictions, which can be represented in SRLS by a rhombic local ordering tensor with components S(0)(2) and S(2)(2). The principal axes frame of this tensor is obviously different from the axial frames of the magnetic tensors. Here, we find that -0.2 ≤ S(0)(2) ≤ 0.5 and -0.4 ≤ S(2)(2) ≤ 0. MF features a single "generalized" order parameter, S, confined to the 0-0.316 range; the local geometry is inherently simple. The parameter S is inaccurate, having absorbed unaccounted for effects, notably S(2)(2) ≠ 0. We find that the methionine methyls (the other methyl types) reorient with rates of 8.6 × 10⁹ to 21.4 × 10⁹ (0.67 × 10⁹ to 6.5 × 10⁹) 1/s. The corresponding activation energies are 10 (10-27) kJ/mol. By contrast, MF yields inaccurate effective local motional correlation times, τ(e), with nonphysical temperature dependence. Thus, the problematic S- and τ(e)-based MF picture of methyl dynamics has been replaced with an insightful physical picture based on a local ordering tensor related to structural features, and a local diffusion tensor that yields accurate activation energies.

Exploring the role of structure and dynamics in the function of chymotrypsin inhibitor 2

Increasing awareness of the possible role of internal dynamics in protein function has led to the development of new methods for experimentally characterizing protein dynamics across multiple time scales, especially using NMR spectroscopy. A few analyses of the conformational dynamics of proteins ranging from nonallosteric single domains to multidomain allosteric enzymes are now available; however, demonstrating a connection between dynamics and function remains difficult on account of the comparative lack of studies examining both changes in dynamics and changes in function in response to the same perturbations. In previous work, we characterized changes in structure and dynamics on the ps–ns time scale resulting from hydrophobic core mutations in chymotrypsin inhibitor 2 and found that there are moderate, persistent global changes in dynamics in the absence of gross structural changes (Whitley et al., Biochemistry 2008;47:8566–8576). Here, we assay those and additional mutants for inhibitory ability toward the serine proteases elastase and chymotrypsin to determine the effects of mutation on function. Results indicate that core mutation has only a subtle effect on CI2 function. Using chemical shifts, we also studied the effect of complex formation on CI2 structure and found that perturbations are greatest at the complex interface but also propagate toward CI2's hydrophobic core. The structure–dynamics–function data set completed here suggests that dynamics plays a limited role in the function of this small model system, although we do observe a correlation between nanosecond-scale reactive loop motions and inhibitory ability for mutations at one key position in the hydrophobic core.

Enhanced stability of a protein with increasing temperature

The unusual stability of a structured but locally flexible protein, human growth hormone (hGH) at pH 2.7, was investigated using the temperature dependence of the nanosecond-picosecond dynamics of the backbone amide groups obtained from (15)N NMR relaxation data. It is found that the flexibility of the backbone of the helices decreases with temperature in the range from 24 °C to ∼40 °C, corresponding to an increasing stability. A concomitant increase with temperature of the electrostatic interactions between charged residues forming an interhelical network of salt bridges at the center of the four-helix core suggests that these interactions give rise to the decreasing flexibility and increasing stability of the protein. However, numerous hydrophobic interactions in the interior of the four-helix core may also contribute. Above ∼40 °C, where the thermal energy overcomes the electrostatic and hydrophobic interactions, a substantial increase in the flexibility of the helix backbones results in a highly positive contribution from the local conformational heat capacity, C(p, conf), of the helix backbones to the total heat capacity, C(p), of the protein. This reduces the change in heat capacity upon unfolding, ΔC(p), increases the change in the Gibbs free energy, ΔG(unfold), and stabilizes the protein at high temperatures. A similar decrease in flexibility is found near other salt bridges in hGH and in Calmodulin and may be of general importance for the thermostability of proteins and, in particular, of the salt bridge intensive thermophilic proteins.

Interpretation of biomolecular NMR spin relaxation parameters

Biomolecular nuclear magnetic resonance (NMR) spin relaxation experiments provide exquisite information on the picosecond to nanosecond timescale motions of bond vectors. Spin-lattice (T1) and spin-spin (T2) relaxation times and the steady-state nuclear Overhauser effect (NOE) are the first set of parameters extracted from typical 15N or 13C NMR relaxation experiments. Therefore, verifying that T1, T2, and NOE are consistent with theoretical predictions is an important step before carrying out the more detailed model-free and reduced spectral density mapping analyses commonly employed. In this mini-review, we discuss the essential motional parameters used to describe biomolecular dynamics in the context of a variety of examples of folded and intrinsically disordered proteins and peptides in aqueous and membrane mimetic environments. Estimates of these parameters can be used as input for an online interface, introduced herein, allowing plotting of trends of T1, T2, and NOE with magnetic field strength. The plots may serve as a first-check to the spectroscopist preparing to embark on a detailed NMR relaxation analysis.

The role of conformational entropy in molecular recognition by calmodulin

The physical basis for high affinity interactions involving proteins is complex and potentially involves a range of energetic contributions. Among these are changes in protein conformational entropy, which cannot yet be reliably computed from molecular structures. We have recently employed changes in conformational dynamics as a proxy for changes in conformational entropy of calmodulin upon association with domains from regulated proteins. The apparent change in conformational entropy was linearly related to the overall binding entropy. This view warrants a more quantitative foundation. Here we calibrate an “entropy meter” employing an experimental dynamical proxy based on NMR relaxation and show that changes in the conformational entropy of calmodulin are a significant component of the energetics of binding. Furthermore, the distribution of motion at the interface between the target domain and calmodulin are surprisingly non-complementary. These observations promote modification of our understanding of the energetics of protein-ligand interactions.

Ligand-induced changes in the structure and dynamics of Escherichia coli peptide deformylase

Peptide deformylase (PDF) is an enzyme that is responsible for removing the formyl group from nascently synthesized polypeptides in bacteria, attracting much attention as a potential target for novel antibacterial agents. Efforts to develop potent inhibitors of the enzyme have progressed on the basis of classical medicinal chemistry, combinatorial chemistry, and structural approaches, yet the validity of PDF as an antibacterial target hangs, in part, on the ability of inhibitors to selectively target this enzyme in favor of structurally related metallohydrolases. We have used (15)N NMR spectroscopy and isothermal titration calorimetry to investigate the high-affinity interaction of EcPDF with actinonin, a naturally occurring potent EcPDF inhibitor. Backbone amide chemical shifts, residual dipolar couplings, hydrogen-deuterium exchange, and (15)N relaxation reveal structural and dynamic effects of ligand binding in the immediate vicinity of the ligand-binding site as well as at remote sites. A comparison of the crystal structures of free and actinonin-bound EcPDF with the solution data suggests that most of the consequences of the ligand binding to the protein are lost or obscured during crystallization. The results of these studies improve our understanding of the thermodynamic global minimum and have important implications for structure-based drug design.

Evidence for domain motion in proteins affecting global diffusion properties: a nuclear magnetic resonance study

The rotational diffusion of proteins is an important hydrodynamic property. Compact protein structures were found previously to exhibit hydration layer viscosity, etaloc, higher than the viscosity of bulk water, eta. This implies an apparent activation energy for rotational diffusion higher than the activation energy of water viscosity, Eeta=15.4+/-0.3 kJ/mol. In this study we examine etaloc of internally mobile proteins using 15N spin relaxation methods. We also examine the activation enthalpy, DeltaH#, and activation entropy, DeltaS#, for rotational diffusion. Of particular relevance are internally mobile ligand-free forms and compact ligand-bound forms of multidomain proteins. Adenylate kinase (AKeco) and Ca2+-calmodulin (Ca2+-CaM) are typical examples. For AKeco (Ca2+-CaM) we find that DeltaH# is 14.5+/-0.5 (15.7+/-0.4) kJ/mol. For the complex of AKeco with the inhibitor AP5A (the complex of Ca2+-CaM with the peptide smMLCKp), we find that DeltaH# is 18.1+/-0.7 (18.2+/-0.5) kJ/mol. The internally mobile outer surface protein A has DeltaH#=12.6+/-0.8 kJ/mol, and the compact protein Staphylococcal nuclease has DeltaH#=18.8+/-0.6 kJ/mol. For the internally mobile and compact proteins studied, <|DeltaS(|> equals 62+/-7 J/(mol K) and 44+/-5 J/(mol K), respectively. The fact is that etaloc>eta (DeltaH#>Eeta) for compact proteins was ascribed previously to electrostatic interactions between surface sites and water rigidifying the hydration layer. We find herein that obliteration of these interactions by domain motion leads to etaloc approximately eta, DeltaH# approximately Eeta, and large activation entropy for internally mobile protein structures.

Re-evaluation of the model-free analysis of fast internal motion in proteins using NMR relaxation

NMR spin relaxation retains a central role in the characterization of the fast internal motion of proteins and their complexes. Knowledge of the distribution and amplitude of the motion of amino acid side chains is critical for the interpretation of the dynamical proxy for the residual conformational entropy of proteins, which can potentially significantly contribute to the entropy of protein function. A popular treatment of NMR relaxation phenomena in macromolecules dissolved in liquids is the so-called model-free approach of Lipari and Szabo. The robustness of the mode-free approach has recently been strongly criticized and the remarkable range and structural context of the internal motion of proteins, characterized by such NMR relaxation techniques, attributed to artifacts arising from the model-free treatment, particularly with respect to the symmetry of the underlying motion. We develop an objective quantification of both spatial and temporal asymmetry of motion and re-examine the foundation of the model-free treatment. Concerns regarding the robustness of the model-free approach to asymmetric motion appear to be generally unwarranted. The generalized order parameter is robustly recovered. The sensitivity of the model-free treatment to asymmetric motion is restricted to the effective correlation time, which is by definition a normalized quantity and not a true time constant and therefore of much less interest in this context. With renewed confidence in the model-free approach, we then examine the microscopic distribution of side chain motion in the complex between calcium-saturated calmodulin and the calmodulin-binding domain of the endothelial nitric oxide synthase. Deuterium relaxation is used to characterize the motion of methyl groups in the complex. A remarkable range of Lipari-Szabo model-free generalized order parameters are seen with little correlation with basic structural parameters such as the depth of burial. These results are contrasted with the homologous complex with the neuronal nitric oxide synthase calmodulin-binding domain, which has distinctly different thermodynamic origins for high affinity binding.

An improved picture of methyl dynamics in proteins from slowly relaxing local structure analysis of 2H spin relaxation

Protein dynamics is intimately related to biological function. Core dynamics is usually studied with 2H spin relaxation of the 13CDH2 group, analyzed traditionally with the model-free (MF) approach. We showed recently that MF is oversimplified in several respects. This includes the assumption that the local motion of the dynamic probe and the global motion of the protein are decoupled, the local geometry is simple, and the local ordering is axially symmetric. Because of these simplifications MF has yielded a puzzling picture where the methyl rotation axis is moving rapidly with amplitudes ranging from nearly complete disorder to nearly complete order in tightly packed protein cores. Our conclusions emerged from applying to methyl dynamics in proteins the slowly relaxing local structure (SRLS) approach of Polimeno and Freed (Polimeno, A.; Freed, J. H. J. Phys. Chem. 1995, 99, 10995-11006.), which can be considered the generalization of MF, with all the simplifications mentioned above removed. The SRLS picture derived here for the B1 immunoglobulin binding domain of peptostreptococcal protein L, studied over the temperature range of 15-45 degrees C, is fundamentally different from the MF picture. Thus, methyl dynamics is characterized structurally by rhombic local potentials with varying symmetries and dynamically by tenfold slower rates of local motion. On average, potential rhombicity decreases, mode-coupling increases, and the rate of local motion increases with increasing temperature. The average activation energy for local motion is 2.0 +/- 0.2 kcal/mol. Mode-coupling affects the analysis even at 15 degrees C. The accuracy of the results is improved by including in the experimental data set relaxation rates associated with rank 2 coherences.